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Answers without the blur. Sign up and see all textbooks for free! Q. 17

Expert-verified Found in: Page 860 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # What is the dot product of the vector functions ${r}_{1}\left(t\right)=\left({x}_{1}\left(t\right),{y}_{1}\left(t\right)\right)\mathrm{and}{r}_{2}\left(t\right)=\left({x}_{2}\left(t\right),{y}_{2}\left(t\right)\right)?$

The dot product of the given two vector functions is ${x}_{1}\left(t\right){x}_{2}\left(t\right)+{y}_{1}\left(t\right){y}_{2}\left(t\right).$

See the step by step solution

## Step 1. Given Information.

The given vector functions are ${r}_{1}\left(t\right)=\left({x}_{1}\left(t\right),{y}_{1}\left(t\right)\right)\mathrm{and}{r}_{2}\left(t\right)=\left({x}_{2}\left(t\right),{y}_{2}\left(t\right)\right).$

## Step 2. Find the dot product.

Let's find the dot product,

${r}_{1}\left(t\right)·{r}_{2}\left(t\right)\phantom{\rule{0ex}{0ex}}=\left({x}_{1}\left(t\right),{y}_{1}\left(t\right)\right)·\left({x}_{2}\left(t\right),{y}_{2}\left(t\right)\right)\phantom{\rule{0ex}{0ex}}=\left({x}_{1}\left(t\right)i+{y}_{1}\left(t\right)j\right)·\left({x}_{2}\left(t\right)i+{y}_{2}\left(t\right)j\right)\phantom{\rule{0ex}{0ex}}={x}_{1}\left(t\right){x}_{2}\left(t\right)\left(i·i\right)+{x}_{1}\left(t\right){y}_{2}\left(t\right)\left(i·j\right)+{y}_{1}\left(t\right){x}_{2}\left(t\right)\left(j·i\right)+{y}_{1}\left(t\right){y}_{2}\left(t\right)\left(j·j\right)\phantom{\rule{0ex}{0ex}}\left[\mathrm{Since}i·i=1,i·j=0,j·i=0,j·j=1\right]\phantom{\rule{0ex}{0ex}}={x}_{1}\left(t\right){x}_{2}\left(t\right)+{y}_{1}\left(t\right){y}_{2}\left(t\right)$

Thus, the product of two vector functions is a scalar function. ### Want to see more solutions like these? 